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Publications and source records attributed to Huishi Li.
Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. Based on the Gröbner basis theory for $A$ and for free modules over $A$, an elimination theory for left ideals of $A$ and an elimination theory for submodules of free $A$-modules are established.
Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. It is shown that if $A$ is an $\mathbb{N}$-graded algebra of $({\cal B},d(~))$-type, then $A$ has a graded monomial ordering $\prec_{gr}$. It is also shown that $A$ is an $\mathbb{N}$-filtered algebra of $({\cal B},d(~))$-type if and only if $A$ has a graded momomial ordering, where ${\cal B}$ is the PBW basis of $A$.
Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a finitely generated $K$-algebra with the PBW $K$-basis ${\cal B}=\{a_{1}^{α_1}\cdots a_{n}^{α_n}~|~(α_1,\ldots ,α_n)\in\mathbb{N}^n\}$. It is shown that if $L$ is a nonzero left ideal of $A$ with GK.dim$(A/L)=d<n$ ($=$ the number of generators of $A$), then $L$ has the {\it elimination property} in the sense that ${\bf V}(U)\cap L\ne \{0\}$ for every subset $U=\{ a_{i_1},\ldots ,a_{i_{d+1}}\}\subset\{a_1,\ldots ,a_n\}$ with $i_1<i_2<\cdots <i_{d+1}$, where ${\bf V}(U)=K$-span$\{a_{i_1}^{α_1}\cdots a_{i_{d+1}}^{α_{d+1}}~|~(α_1,\ldots ,α_{d+1})\in\mathbb{N}^{d+1}\}$. In terms of the structural properties of $A$, it is also explored when the condition GK.dim$(A/L)<n$ may hold for a left ideal $L$ of $A$. Moreover, from the viewpoint of realizing the elimination property by means of Gröbner bases, it is demonstrated that if $A$ is in the class of binomial skew polynomial rings [G-I2, Serdica Math. J., 30(2004)] or in the class of solvable polynomial algebras [K-RW, J. Symbolic Comput., 9(1990)], then every nonzero left ideal $L$ of $A$ satisfies GK.dim$(A/L)<$ GK.dim$A=n$ ($=$ the number of generators of $A$), thereby $L$ has the elimination property.
This text consists of five relatively systematic notes on Gröbner bases and free resolutions of modules over solvable polynomial algebras.
Let $\KX =K\langle X_1,\ldots ,X_n\rangle$ be the free algebra generated by $X=\{ X_1,\ldots ,X_n\}$ over a field $K$. It is shown that with respect to any weighted $\mathbb{N}$-gradation attached to $\KX$, minimal homogeneous generating sets for finitely generated graded (two-sided) ideals of $\KX$ can be algorithmically computed, and that if an ungraded (two-sided) ideal $I$ of $\KX$ has a finite Gröbner basis $\G$ with respect to a graded monomial ordering on $\KX$, then a minimal standard basis for $I$ can be computed via computing a minimal homogeneous generating set of the associated graded ideal $\langle\LH (I)\rangle$.
It is shown that the methods and algorithms, developed in (A. Capani et al., Computing minimal finite free resolutions, {\it Journal of Pure and Applied Algebra}, (117& 118)(1997), 105 -- 117; M. Kreuzer and L. Robbiano, {\it Computational Commutative Algebra 2}, Springer, 2005.) for computing minimal homogeneous generating sets of graded submodules and graded quotient modules of free modules over a commutative polynomial algebra, can be adapted for computing minimal homogeneous generating sets of graded submodules and graded quotient modules of free modules over a weighted $\mathbb{N}$-graded solvable polynomial algebra, where solvable polynomial algebras are in the sense of (A. Kandri-Rody and V. Weispfenning, Non-commutative Gröbner bases in algebras of solvable type. {\it J. Symbolic Comput.}, 9(1990), 1--26). Consequently, algorithmic procedures for computing minimal finite graded free resolutions over weighted $\mathbb{N}$-graded solvable polynomial algebras are achieved.
Let $A=K[a_1,\ldots,a_n]$ be a weighted $\mathbb{N}$-filtered solvable polynomial algebra with filtration $FA=\{ F_pA\}_{p\in\mathbb{N}}$, where solvable polynomial algebras are in the sense of (A. Kandri-Rody and V. Weispfenning, Non-commutative Gröbner bases in algebras of solvable type. {\it J. Symbolic Comput.}, 9(1990), 1--26), and $FA$ is constructed with respect to a positive-degree function $d(~)$ on $A$. By introducing minimal F-bases and minimal standard bases respectively for left $A$-modules and their submodules with respect to good filtrations, minimal filtered free resolutions for finitely generated $A$-modules are introduced. It is shown that any two minimal F-bases, respectively any two minimal standard bases have the same number of elements and the same number of elements of the same filtered degree; that minimal filtered free resolutions are unique up to strict filtered isomorphism of chain complexes in the category of filtered $A$-modules; and that minimal finite filtered free resolutions can be algorithmically computed by employing Gröbner basis theory for modules over $A$ with respect to any graded left monomial ordering on free left $A$-modules.
For the solvable polynomial algebras introduced and studied by Kandri-Rody and Weispfenning [J. Symbolic Comput., 9(1990)], a constructive characterization is given in terms of Gröbner bases for ideals of free algebras, thereby solvable polynomial algebras are completely determinable in a computational way.
Let $Λ$ be a commutative Noetherian ring, and let $I$ be a proper ideal of $Λ$, $R=Λ/I$. Consider the polynomial rings $T=Λ[x_1,...x_n]$ and $A=R[x_1,...,x_n]$. Suppose that linear equations are solvable in $Λ$. It is shown that linear equations are solvable in $R$ (thereby theoretically Gröbner bases for ideals of $A$ are well defined and constructible) and that practically Gröbner bases in $A$ with respect to any given monomial ordering can be obtained by constructing Gröbner bases in $T$, and moreover, all basic applications of a Gröbner basis at the level of $A$ can be realized by a Gröbner basis at the level of $T$. Typical applications of this result are demonstrated respectively in the cases where $Λ=D$ is a PID, $Λ=D[y_1,...,y_m]$ is a polynomial ring over a PID $D$, and $Λ=K[y_1,...,y_m]$ is a polynomial ring over a field $K$.
Let $K =K $ be the free $K$-algebra on $X={X_1,...,X_n}$ over a field $K$, which is equipped with a weight $\mathbb{N}$-gradation (i.e., each $X_i$ is assigned a positive degree), and let ${\cal G}$ be a finite homogeneous Gröbner basis for the ideal $I=<{\cal G}>$ of $K $ with respect to some monomial ordering $\prec$ on $K $. It is proved that if the monomial algebra $K /<{\bf LM}({\cal G})>$ is semi-prime, where ${\bf LM}({\cal G})$ is the set of leading monomials of ${\cal G}$ with respect to $\prec$, then the $\mathbb{N}$-graded algebra $A=K /I$ is semiprimitive (in the sense of Jacobson). In the case that ${\cal G}$ is a finite non-homogeneous Gröbner basis with respect to a graded monomial ordering $\prec_{gr}$, and the $\mathbb{N}$-filtration $FA$ of the algebra $A=K /I$ induced by the $\mathbb{N}$-grading filtration $FK $ of $K $ is considered, if the monomial algebra $K /<{\bf LM}({\cal G})>$ is semi-prime, then it is proved that the associated $\mathbb{N}$-graded algebra $G(A)$ and the Rees algebra $\widetilde{A}$ of $A$ determined by $FA$ are all semiprimitive.
Let $Γ$ be a cancelation monoid with the neutral element $e$. Consider a $Γ$-graded ring $A=\oplus_{γ\inΓ}A_γ$, which is not necessarily commutative. It is proved that $A_e$, the degree-$e$ part of $A$, is a local ring in the classical sense if and only if the graded two-sided ideal $\mathfrak{M}$ of $A$ generated by all non-invertible homogeneous elements is a proper ideal. Defining a $Γ$-graded local ring $A$ in terms of this equivalence, it is proved that any two minimal homogeneous generating sets of a finitely generated $Γ$-graded $A$-module have the same number of generators, and furthermore, that most of the basic homological properties of the local ring $A_e$ hold true for $A$ (at least) in the $Γ$-graded context.
Let $K$ be a field, $\mathcal {O}_v$ a valuation ring of $K$ associated to a valuation $v$: $K\rightarrowΓ\cup\{\infty\}$, and ${\bf m}_v$ the unique maximal ideal of $\mathcal {O}_v$. Consider an ideal $\mathcal {I}$ of the free $K$-algebra $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ on $X_1,...,X_n$. If ${\cal I}$ is generated by a subset $\mathcal {G}\subset{\cal O}_v\langle X\rangle$ which is a monic Gröbner basis of ${\cal I}$ in $K\langle X\rangle$, where $\mathcal {O}_v\langle X\rangle =\mathcal{O}_v\langle X_1,...,X_n\rangle$ is the free $\mathcal{O}_v$-algebra on $X_1,...,X_n$, then the valuation $v$ induces naturally an exhaustive and separated $Γ$-filtration $F^vA$ for the $K$-algebra $A=K\langle X\rangle /\mathcal {I}$, and moreover $\mathcal{I}\cap\mathcal{O}_v\langle X\rangle =\langle\mathcal{G}\rangle$ holds in $\mathcal{O}_v\langle X\rangle$; it follows that, if furthermore $\mathcal{G}\not\subset {\bf m}_v{O}_v\langle X\rangle$ and $k\langle X\rangle /\langle\overline{\mathcal G}\rangle$ is a domain, where $k=\mathcal{O}_v/{\bf m}_v$ is the residue field of $\mathcal{O}_v$, $k\langle X\rangle =k\langle X_1,...,X_n\rangle$ is the free $k$-algebra on $X_1,...,X_n$, and $\overline{\mathcal G}$ is the image of $\mathcal{G}$ under the canonical epimorphism $\mathcal{O}_v\langle X\rangle\rightarrow k\langle X\rangle$, then $F^vA$ determines a valuation function $A\rightarrow Γ\cup\{\infty\}$, and thereby $v$ extends naturally to a valuation function on the (skew-)field $Δ$ of fractions of $A$ provided $Δ$ exists.
Let $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ be the free algebra of $n$ generators over a field $K$, and let $R\langle X\rangle =R\langle X_1,...,X_n\rangle$ be the free algebra of $n$ generators over an arbitrary commutative ring $R$. In this semi-expository paper, it is clarified that any monic Gröbner basis in $K\langle X\rangle$ may give rise to a monic Gröbner basis of the same type in $R\langle X\rangle$, and vice versa. This fact turns out that many important $R$-algebras have defining relations which form a monic Gröbner basis, and consequently, such $R$-algebras may be studied via a nice PBW structure theory as that developed for quotient algebras of $K\langle X\rangle$ in ([LWZ], [Li2, 3]).
Let $K$ be a field and $R=\oplus_{p\in\mathbb{N}}R_p$ an $\mathbb{N}$-graded $K$-algebra, which has an SM $K$-basis (i.e. a skew multiplicative $K$-basis) such that $R$ holds a Gröbner basis theory. It is proved that there is a one-to-one correspondence between the set of Gröbner bases in $R$ and the set of dh-closed homogeneous Gröbner bases in the polynomial algebra $R[t]$; and that the similar result holds true if $R$ and $R[t]$ are replaced respectively by the free algebra $K< X_1,...,X_n>$ and the free algebra $K< X_1,...,X_n,T>$. Moreover, it is shown that dh-closed graded ideals in $R[t]$ and $K< X_1,...,X_n, T>$ can be realized by dh-closed homogeneous Gröbner bases. The latter result indeed tells us that algebras defined by dh-homogeneous Gröbner bases can be studied as Rees algebras effectively via more simpler algebras as demonstrated in ([7], [8]).
In this paper, we introduce (almost) skew 2-nomial algebras and look for a one-sided or two-sided Gröbner basis theory for such algebras at a modest level. That is, we establish the existence of a skew multiplicative $K$-basis for every skew 2-nomial algebra, and we explore the existence of a (left, right, or two-sided) monomial ordering for an (almost) skew 2-nomial algebra. As distinct from commonly recognized algebras holding a Gröbner basis theory (such as algebras of the solvable type [K-RW] and some of their homomorphic images), a subclass of skew 2-nomial algebras that have a left Gröbner basis theory but may not necessarily have a two-sided Gröbner basis theory, respectively a subclass of skew 2-nomial algebras that have a right Gröbner basis theory but may not necessarily have a two-sided Gröbner basis theory, are determined such that numerous quantum binomial algebras (which provide binomial solutions to the Yang-baxter equation [Laf], [G-I2]) are involved.
By employing the (de)homogenization technique in a relatively extensive setting, this note studies in detail the relation between non-homogeneous Gröbner bases and homogeneous Gröbner bases. As a consequence, a general principle of computing Gröbner bases (for an ideal and its homogenization ideal) by passing to homogenized generators is clarified systematically. The obtained results improve and strengthen the work of [LWZ], [Li1], [Li2], [Li3], and very recent [SL] concerning the same topic.
Let $A=K< X_1,...,X_n> /< {\cal G}>$ be a $K$-algebra defined by a finite Gröbner basis ${\cal G}$. It is shown how to use the Ufnarovski graph $Γ({\bf LM}({\cal G}))$ and the graph of $n$-chains $Γ_{\rm C}({\bf LM}({\cal G}))$ to calculate gl.dim$G^{\mathbb{N}}(A)$ and gl.dim$\widetilde{A}$, where $G^{\mathbb{N}}(A)$, respectively $\widetilde{A}$, is the associated $\mathbb{N}$-graded algebra of $A$, respectively the Rees algebra of $A$ with respect to the $\mathbb{N}$-filtration $FA$ of $A$ induced by a weight $\mathbb{N}$-grading filtration of $K< X_1,...,X_n>$.
Let $K$ be an infinite field and $K< X> =K< X_1,...,X_n>$ the free associative algebra generated by $X=\{X_1,...,X_n\}$ over $K$. It is proved that if $I$ is a two-sided ideal of $K< X>$ such that the $K$-algebra $A=K< X> /I$ is almost commutative in the sense of [3], namely, with respect to its standard $\mathbb{N}$-filtration $FA$, the associated $\mathbb{N}$-graded algebra $G(A)$ is commutative, then $I$ is generated by a finite Gröbner basis. Therefor, every quotient algebra of the enveloping algebra $U(\mathbf{g})$ of a finite dimensional $K$-Lie algebra $\mathbf{g}$ is, as a noncommutative algebra of the form $A=K< X> /I$, defined by a finite Gröbner basis in $K< X>$.